## Table of contents

- Locally flat chains in Euclidean space
- Differentiability theory for distributions and subsets of Euclidean space
- Survey on varifolds suitable for graduate students
- Partial differential equations on singular surfaces with mean curvature – foundations
- Regularity of singular surfaces with mean curvature
- Basic geometric properties of singular surfaces with mean curvature
- Conference proceedings
- List of minor corrections

This page was last modified on: 06 July 2024, 11:55, CST

## Locally flat chains in Euclidean space

U. Menne, C. Scharrer

[15] *A priori bounds for geodesic diameter. Part I.
Integral chains with coefficients in a complete normed
commutative group*, 41 pages.

Rev. Mat. Iberoam., accepted, Apr 2024.

ArXiv: 2206.14046v2 [math.DG].

**Abstract:** As service to the community, we
provide—for Euclidean space—a basic treatment of locally
rectifiable chains and of the complex of locally integral
chains. In this setting, we may beneficially develop the idea
of a complete normed commutative group bundle over the
Grassmann manifold whose fibre is the coefficient group of the
chains. Our exposition also sheds new light on some algebraic
aspects of the theory. Finally, we indicate an extension to a
geometric approach to locally flat chains centring on locally
rectifiable chains rather than completion procedures.

## Differentiability theory for distributions and subsets of Euclidean space

U. Menne

[14] *Pointwise differentiability of higher-order for
distributions*, 32 pages.

Anal. PDE, 14(2):323–354, Mar 2021.

DOI: 10.2140/apde.2021.14.323.
ArXiv: 1803.10855v2 [math.FA].

**Abstract:** For distributions, we build a
theory of higher order pointwise differentiability comprising,
for order zero, Łojasiewicz's notion of point value.
Results include Borel regularity of differentials, higher
order rectifiability of the associated jets, a
Rademacher-Stepanov type differentiability theorem, and a
Lusin type approximation. A substantial part of this
development is new also for zeroth order. Moreover, we
establish a Poincaré inequality involving the natural
norms of negative order of differentiability. As a corollary,
we characterise pointwise differentiability in terms of point
values of distributional partial derivatives.

U. Menne, M. Santilli

[11] * A geometric second-order-rectifiable stratification
for closed subsets of Euclidean space*, 14 pages.

Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 19(3):1185–1198,
Sep 2019.

DOI: 10.2422/2036-2145.201703_021. ArXiv: 1703.09561v2 [math.CA].

**Abstract:** Defining the *m*-th stratum of
a closed subset of an *n* dimensional Euclidean space to
consist of those points, where it can be touched by a ball
from at least *n−m* linearly independent directions, we
establish that the *m*-th stratum is second-order
rectifiable of dimension *m* and a Borel set. This was
known for convex sets, but is new even for sets of positive
reach. The result is based on a sufficient condition of
parametric type for second-order rectifiability.

U. Menne

[9] *Pointwise differentiability of higher order for
sets*, 31 pages.

Ann. Global Anal. Geom., 55(3), 591–621, Apr 2019.

Springer Nature SharedIt: https://rdcu.be/bgUqa

DOI: 10.1007/s10455-018-9642-0. ArXiv: 1603.08587v2 [math.DG].

**Abstract:** The present paper develops two
concepts of pointwise differentiability of higher order for
arbitrary subsets of Euclidean space defined by comparing
their distance functions to those of smooth submanifolds.
Results include that differentials are Borel functions, higher
order rectifiability of the set of differentiability points,
and a Rademacher result. One concept is characterised by a
limit procedure involving inhomogeneously dilated sets. The
original motivation to formulate the concepts stems from
studying the support of stationary integral varifolds. In
particular, strong pointwise differentiability of every
positive integer order is shown at almost all points of the
intersection of the support with a given plane.

## Survey on varifolds suitable for graduate students

U. Menne

[12] *The concept of varifold*, 5 pages.

Notices Amer. Math. Soc., 64(10):1148–1152, Nov 2017.

DOI: 10.1090/noti1589. ArXiv: 1705.05253v4 [math.DG].

**Abstract:** We survey – by means of 20 examples
– the concept of varifold, as generalised submanifold, with
emphasis on regularity of integral varifolds with mean
curvature, while keeping prerequisites to a minimum. Integral
varifolds are the natural language for studying the
variational theory of the area integrand if one considers, for
instance, existence or regularity of stationary (or, stable)
surfaces of dimension at least three, or the limiting
behaviour of sequences of smooth submanifolds under area and
mean curvature bounds.

## Partial differential equations on singular surfaces with mean curvature – foundations

U. Menne

[17] *A sharp lower bound on the mean curvature integral
with critical power for integral varifolds*, **144
pages**.

Manuscript dated 2014. Complete proofs, few reader guidance,
24 chapters, 1 appendix, 144 pages. Part I (Chapters 1-14) is
published in [6]; Chapter 15 is published as
part of [8]; some material of Chapter 19 is
included in [9]; other chapters shall be
generalised (with collaborators) and published elsewhere.

ArXiv: 2310.01754v1 [math.DG], Oct 2023.

**Abstract:**

Part I Weakly differentiable functions

Part II PDEs on varifolds

15 Sobolev spaces … 75

16 Preliminaries … 82

17 Maximum estimates … 86

18 Second order flatness of Lebesgue spaces for integral
varifolds with subcritical integrability of the mean curvature
… 94

19 Second order differentiability of the support of integral
varifolds with critical integrability of the mean curvature
… 100

20 Harnack inequality … 108

Part III The generalised Gauss map

21 Points of finite lower density … 118

22 Points of infinite density … 130

23 Area formula for the generalised Gauss map … 137

24 A sharp lower bound on the mean curvature integral with
critical power for integral varifolds … 138

Appendix

A Monotonicity identity … 139

U. Menne

[8] *Sobolev functions on varifolds*, **50
pages**.

Proc. Lond. Math. Soc. (3), 113(6):725–774, Dec 2016.

Wiley Content Sharing: https://rdcu.be/bV3Rw

DOI: 10.1112/plms/pdw023. ArXiv: 1509.01178v3 [math.CA].

**Abstract:** This paper introduces first-order
Sobolev spaces on certain rectifiable varifolds. These
complete locally convex spaces are contained in the generally
non-linear class of generalised weakly differentiable
functions and share key functional analytic properties with
their Euclidean counterparts.

Assuming the varifold to satisfy a uniform lower density bound
and a dimensionally critical summability condition on its mean
curvature, the following statements hold. Firstly, continuous
and compact embeddings of Sobolev spaces into Lebesgue spaces
and spaces of continuous functions are available. Secondly,
the geodesic distance associated to the varifold is a
continuous, not necessarily Hölder continuous Sobolev function
with bounded derivative. Thirdly, if the varifold additionally
has bounded mean curvature and finite measure, then the
present Sobolev spaces are isomorphic to those previously
available for finite Radon measures yielding many new results
for those classes as well.

Suitable versions of the embedding results obtained for
Sobolev functions hold in the larger class of generalised
weakly differentiable functions.

U. Menne

[6] *Weakly differentiable functions on varifolds*,
**112 pages**.

Indiana Univ. Math. J., 65(3):977–1088, Jul 2016.

DOI: 10.1512/iumj.2016.65.5829. ArXiv: 1411.3287v1 [math.DG].

**Abstract:** The present paper is intended to
provide the basis for the study of weakly differentiable
functions on rectifiable varifolds with locally bounded first
variation. The concept proposed here is defined by means of
integration-by-parts identities for certain compositions with
smooth functions. In this class, the idea of zero boundary
values is realised using the relative perimeter of superlevel
sets. Results include a variety of Sobolev Poincaré-type
embeddings, embeddings into spaces of continuous and sometimes
Hölder-continuous functions, and pointwise differentiability
results both of approximate and integral type as well as
coarea formulae.

As a prerequisite for this study, decomposition properties of
such varifolds and a relative isoperimetric inequality are
established. Both involve a concept of distributional boundary
of a set introduced for this purpose.

As applications, the finiteness of the geodesic distance
associated with varifolds with suitable summability of the
mean curvature and a characterisation of curvature varifolds
are obtained.

## Regularity of singular surfaces with mean curvature

S. Kolasiński, U. Menne

[7] *Decay rates for the quadratic and super-quadratic
tilt-excess of integral varifolds*, **56
pages**.

NoDEA Nonlinear Differential Equations Appl., 24:Art. 17, 56,
Mar 2017.

DOI: 10.1007/s00030-017-0436-z. ArXiv: 1501.07037v2 [math.DG].

**Abstract:** This paper concerns integral
varifolds of arbitrary dimension in an open subset of
Euclidean space satisfying integrability conditions on their
first variation. Firstly, the study of pointwise power decay
rates almost everywhere of the quadratic tilt-excess is
completed by establishing the precise decay rate for
two-dimensional integral varifolds of locally bounded first
variation. In order to obtain the exact decay rate, a coercive
estimate involving a height-excess quantity measured in Orlicz
spaces is established. Moreover, counter-examples to pointwise
power decay rates almost everywhere of the super-quadratic
tilt-excess are obtained. These examples are optimal in terms
of the dimension of the varifold and the exponent of the
integrability condition in most cases, for example if the
varifold is not two-dimensional. These examples also
demonstrate that within the scale of Lebesgue spaces no local
higher integrability of the second fundamental form, of an at
least two-dimensional curvature varifold, may be deduced from
boundedness of its generalised mean curvature vector. Amongst
the tools are Cartesian products of curvature varifolds.

U. Menne

[4] *Second order rectifiability of integral varifolds of
locally bounded first variation*, **55
pages**.

J. Geom. Anal., 23(2):709–763, Apr 2013.

DOI: 10.1007/s12220-011-9261-5. ArXiv: 0808.3665v3 [math.DG].

**Abstract:** It is shown that every integral
varifold in an open subset of Euclidean space whose first
variation with respect to area is representable by integration
can be covered by a countable collection of submanifolds of
the same dimension of class 2 and that their mean curvature
agrees almost everywhere with the variationally defined
generalized mean curvature of the varifold.

U. Menne

[3] *Decay estimates for the quadratic tilt-excess of
integral varifolds*, **83 pages**.

Arch. Ration. Mech. Anal., 204(1):1–83, Apr 2012.

DOI: 10.1007/s00205-011-0468-1. ArXiv: 0909.3253v3 [math.DG].

**Abstract:** This paper concerns integral
varifolds of arbitrary dimension in an open subset of
Euclidean space with their first variation given by either a
Radon measure or a function in some Lebesgue space. Pointwise
decay results for the quadratic tilt-excess are established
for those varifolds. The results are optimal in terms of the
dimension of the varifold and the exponent of the Lebesgue
space in most cases, for example if the varifold is not
two-dimensional.

## Basic geometric properties of singular surfaces with mean curvature

U. Menne, C. Scharrer

[16] *A priori bounds for geodesic diameter. Part II. Fine
connectedness properties of varifolds*, 58 pages.

ArXiv: 2209.05955v2 [math.DG], Aug 2023.

**Abstract:** For varifolds whose first variation
is representable by integration, we introduce the notion of
indecomposability with respect to locally Lipschitzian real
valued functions. Unlike indecomposability, this weaker
connectedness property is inherited by varifolds associated
with solutions to geometric variational problems phrased in
terms of sets, *G* chains, and immersions; yet it is
strong enough for the subsequent deduction of substantial
geometric consequences therefrom. Our present study is based
on several further concepts for varifolds put forward in this
paper: real valued functions of generalised bounded variation
thereon, partitions thereof in general, partition thereof
along a real valued generalised weakly differentiable function
in particular, and local finiteness of decompositions.

U. Menne, C. Scharrer

[13] *A priori bounds for geodesic diameter. Part III. A
Sobolev-Poincaré inequality and applications to a variety of
geometric variational problems*, 46 pages

ArXiv: 1709.05504v2 [math.DG], Mar 2024.

**Abstract:** Based on a novel type of
Sobolev-Poincaré inequality (for generalised weakly
differentiable functions on varifolds), we establish a finite
upper bound of the geodesic diameter of generalised compact
connected surfaces-with-boundary of arbitrary dimension in
Euclidean space in terms of the mean curvatures of the surface
and its boundary. Our varifold setting includes smooth
immersions, surfaces with finite Willmore energy, two-convex
hypersurfaces in level-set mean curvature flow, integral
currents with prescribed mean curvature vector, area
minimising integral chains with coefficients in a complete
normed commutative group, varifold solutions to Plateau’s
problem furnished by min-max methods or by Brakke flow, and
compact sets solving Plateau problems based on Čech homology.
Due to the generally inevitable presence of singularities,
path-connectedness was previously known neither for the class
of varifolds (even in the absence of boundary) nor for the
solutions to the Plateau problems considered.

U. Menne, C. Scharrer

[10] *An isoperimetric inequality for diffused surfaces*,
16 pages.

Kodai Math. J., 41(1):70–85, Mar 2018.

DOI: 10.2996/kmj/1521424824. ArXiv: 1612.03823v2 [math.DG].

**Abstract:** For general varifolds in Euclidean
space, we prove an isoperimetric inequality, adapt the basic
theory of generalised weakly differentiable functions, and
obtain several Sobolev type inequalities. We thereby intend to
facilitate the use of varifold theory in the study of diffused
surfaces.

U. Menne

[2] *A Sobolev Poincaré type inequality for integral
varifolds*, 40 pages.

Calc. Var. Partial Differential Equations, 38(3-4):369–408,
Jul 2010.

DOI: 10.1007/s00526-009-0291-9. ArXiv: 0808.3660v2 [math.DG].

**Abstract:** In this work a local inequality is
provided which bounds the distance of an integral varifold
from a multivalued plane (height) by its tilt and mean
curvature. The bounds obtained for the exponents of the
Lebesgue spaces involved are shown to be sharp.

U. Menne

[1] *Some applications of the isoperimetric inequality for
integral varifolds*, 23 pages.

Adv. Calc. Var., 2(3):247–269, Jul 2009.

DOI: 10.1515/ACV.2009.010. ArXiv: 0808.3652v1 [math.DG].

**Abstract: **In this work the isoperimetric
inequality for integral varifolds of locally bounded first
variation is used to obtain sharp estimates for the size of
the set where the density quotient is small and to generalise
Calderón's and Zygmund's theory of first order
differentiability for functions in Lebesgue spaces from
Lebesgue measure to integral varifolds.

## Conference proceedings

U. Menne

[5] *A sharp lower bound on the mean curvature integral with
critical power for integral varifolds*, 3 pages.

In abstracts from the workshop held July 22-28, 2012,
Organized by Camillo De Lellis, Gerhard Huisken and Robert
Jerrard, Oberwolfach Reports. Vol. 9, no. 3, 2012.

DOI: 10.4171/OWR/2012/36.